A hinged plate equation and iterated Dirichlet Laplace operator on domains with concave corners

By S. Nazarov, Guido Sweers. 2007

67 citations recorded by the source corpus.

Abstract Fourth order hinged plate type problems are usually solved via a system of two second order equations. For smooth domains such an approach can be justified. However, when the domain has a concave corner the bi-Laplace problem with Navier boundary conditions may have two different types of solutions, namely u 1 with u 1 , Δ u 1 ∈ H ˚ 1 and u 2 ∈ H 2 ∩ H ˚ 1 . We will compare these two solutions. A striking difference is that in general only the first solution, obtained by decoupling into a system, preserves positivity, that is, a positive source implies that the solution is positive. The other type of solution is more relevant in the context of the hinged plate. We will also address the higher-dimensional case. Our main analytical tools will be the weighted Sobolev spaces that originate from Kondratiev. In two dimensions we will show an alternative that uses conformal transformation. Next to rigorous proofs the results are illustrated by some numerical experiments for planar domains.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/08f69b699729994658b0866fb0a1abe33f68b0c4

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In Applied Mathematics

In Computational Theory and Mathematics

In Mathematical Physics

Citations in the corpus

The First Biharmonic Steklov Eigenvalue: Positivity Preserving and Shape Optimization (2011, unknown): “This result appears somehow sharp since, in view of [23], we believe that a minimizer might not exist in domains with a concave corner.”
On Positivity for the Biharmonic Operator under Steklov Boundary Conditions (2008, unknown): “For nonsmooth domains not only uν might not be well-defined but if the domain has a reentrant corner then by [24] one knows that the H1( ) × H1( )-solution of a system not necessarily coincides with the H2( )-solution for the original fourth-order problem.”
An Isoperimetric Inequality for Fundamental Tones of Free Plates (2010, background): “Other boundary conditions exist for fourth-order problems, such as the hinged plate investigated by Nazarov and Sweers [13].”
On the first eigenvalue of a fourth order Steklov problem (2009, unknown): “However, in view of [ 19 ], we believe that a minimizer might not exist in domains with a concave corner.”
Positivity preserving property for a class of biharmonic elliptic problems (2006, background): “In this case, the positivity property is not sensitive to the geometric or topological properties of the domain Ω. In fact, smoothness of the domain is here crucial since it has recently been shown by Nazarov and Sweers [ 21 ] that for planar domains with an interior corner, (2) may not hold.”